Final answer:
To find the total differential dz of the function z = -5x² + 4xy, partial derivatives are taken with respect to x and y, yielding dz = (-10x + 4y)dx + (4x)dy.
Step-by-step explanation:
To find the total differential dz of the function z = -5x² + 4xy, we need to use partial derivatives. The total differential dz is given by the formula dz = (∂z/∂x)dx + (∂z/∂y)dy, where ∂ denotes the partial derivative.
First, let's find the partial derivatives of z with respect to x and y. The partial derivative of z with respect to x is ∂z/∂x = -10x + 4y. The partial derivative of z with respect to y is ∂z/∂y = 4x.
Substituting these into the formula for dz, we get:
dz = (-10x + 4y)dx + (4x)dy
This is the full expression for the total differential dz of the given function.